categories: thirding

Let S and C be endo-functors that commute, that is, there's a
natural equivalence c:CS --> SC. If f: F -> SF is a final
S-coalgebra then there a special map CF --> F, to wit, the induced
Cf c_F
coalgebra map from the S-coalgebra CF --> CSF --> SCF.
Now specialize to the case that SX is X*X for an associative
bifunctor and CX is X*X*X. Indeed, specialize further to the
case that * is the ordered-wedge functor so that F is the closed
interval, I. In this special case the induced map I v I v I --> I
is an isomorphism and its inverse makes I a final cubical coalgebra.
I don't know a general theorem that specializes to this result.
Clearly I v I v I --> I can be used to obtain the thirding map
on the interval that I needed for my definition of derivatives.
And clearly there's nothing special about the number three. We obtain
a special isomorphism from every iterated ordered-wedge of I back
to I.
In that previous posting on derivatives I wrote:
Using that the closed interval is the final coalgebra for X v X v X
we can define the thirding map t:I --> I in a manner similar to
(and simpler than) the definition of the halving map.
The problem, of course, was that there was no control on _which_
final cubical coalgebra structure was to be used.